Abstract

Using the integral transformation and inequalities technique, new oscillation criteria are established for fractional differential equations with mixed nonlinearities involving Riemann-Liouville and Caputo fractional derivatives, which generalize and improve some recent results in literature.

MSC:34A08, 34C10.

Keywords

1 Introduction

Fractional differential equations appear more and more frequently in various research areas, such as in modeling mechanical and electrical properties of real materials, as well as in rheological theory and other physical problems, etc.; see, e.g., [1–6]. Differential equations involving the Riemann-Liouville, Caputo, and Grünwald-Letnikov differential operators of fractional order 0<q<1 appear to be important in a number of works, especially in the theory of viscoelasticity and in hereditary solid mechanics.

In [3], the authors obtained new oscillation criteria for a fractional differential equations of the form

Dtqax+f1(t,x)=v(t)+f2(t,x),limt→a+Ja1−qx(t)=a1,

(1)

where the functions f1, f2, and v are continuous.

In this paper, we consider the oscillation theory for a fractional differential equation with mixed nonlinearities of the type

Dtqax−p(t)x(t)+∑i=1mqi(t)|x(t)|λi−1x(t)=v(t),limt→a+Ja1−qx(t)=a1,

(2)

where {p(t)}, {v(t)}, and {qi(t)} (1≤i≤m) are continuous functions on [a,+∞), and λi (1≤i≤m) are ratios of odd positive integers with λ1>⋯>λl>1>λl+1>⋯>λm.

By a solution of equation (2) we mean a function x(t) which is defined for t≥a and satisfies equation (2). Such a solution is said to be oscillatory if it has arbitrarily large zeros on [a,∞); otherwise, it is called nonoscillatory. Equation (2) is said to be oscillatory if all its solutions are oscillatory.

By Dtqa we denote the Riemann-Liouville differential operator of order q with 0<q≤1. For p≥0, the operator Jap defined by

Japx(t)=1Γ(p)∫at(t−s)p−1x(s)ds,Ja0x=x

(3)

is called the Riemann-Liouville fractional integral operator. The Riemann-Liouville differential operator Dtqa of order q for 0<q≤1 is defined by Dtqax(t)=ddtJa1−qx(t) and, more generally, if m≥1 is an integer and m−1<q≤m, then

Dtqax(t)=dmdtmJam−qx(t).

(4)

In [[2], Lemma 5.3], under much weaker assumptions on p(t), v(t) and qi(t), the initial value problem (2) is equivalent to the Volterra fractional integral equation

Therefore, a function x(t) is a solution of (5) if and only if it is a solution of fractional differential equation (2).

In this paper, using the similar methods as that in [7], we give new oscillation criteria for equation (2) which generalize and improve the main results in paper [3] and references cited therein. Examples are given to each of these equations.

where K=max{λ1−1,maxl+1≤i≤m(1−λi)(λiA)λi1−λi}. Note that the improper integral on the right is divergent. Taking the limit inferior of both sides of inequality (13) as t→∞, we get a contradiction to condition (9). In the case x(t) is eventually negative, a similar argument leads to a contradiction to (10). This completes the proof of Theorem 2.1. □

Following the proof of Theorem 2.1, we can easily obtain the following corollaries.

Corollary 2.1Supposep(t)>0, qi(t)≥0, 1≤i≤m. If (9), (10) hold for some constantK1>0, then equation (2) is oscillatory.

Proof Suppose to the contrary that there exists a nonoscillatory solution x(t) of equation (2). Without loss of generality, we may suppose that x(t) is an ultimately positive solution of equation (2). So, there exists T>a such that x(t)>0 for t≥T. It follows from equation (2) that

Using similar methods, the oscillation criteria can be obtained for Caputo’s case.

Theorem 3.1Assume that condition (8) holds. If

lim inft→∞t1−m∫at(t−s)q−1(v(s)+K∑i=1mpλiλi−1(s)|qi(s)|11−λi)ds=−∞

(17)

and

lim supt→∞t1−m∫at(t−s)q−1(v(s)+K∑i=1mpλiλi−1(s)|qi(s)|11−λi)ds=∞

(18)

for some constantK>0, then every solution of equation (16) is oscillatory.

Corollary 3.1Supposep(t)>0, qi(t)≥0, 1≤i≤m. If (17), (18) hold for some constantK1>0, then equation (16) is oscillatory.

Corollary 3.2Supposep(t)>0, qi(t)≤0, 1≤i≤m. If (17), (18) hold for some constantK2>0, then equation (16) is oscillatory.

Corollary 3.3If condition (14) holds, and there exists a positive functionr(t)on[a,∞)such that

lim inft→∞t1−m∫at(t−s)q−1[v(s)+K3∑i=1mrλiλi−1(s)|qi(s)|11−λi]ds=−∞

and

lim supt→∞t1−m∫at(t−s)q−1[v(s)+K3∑i=1mrλiλi−1(s)|qi(s)|11−λi]ds=∞

for some constantK3>0, then every solution of equation (16) is oscillatory.

Remark 3.1 In [1], the Grünwald-Letnikov fractional derivative, under the assumption that the function x(t) must be m+1 times continuously differentiable, can be obtained from (4) under the same assumption by performing repeatedly integration by parts and differentiation. Therefore, our results are suitable for the Grünwald-Letnikov fractional derivative approaches, too.

4 Examples

In this section, we give the following examples to illustrate the effectiveness of our theorems.

we get that neither (9) nor (10) is satisfied. We can also easily verify that x(t)=t is a nonoscillatory solution of (20).

Declarations

Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present article. This project is supported by the NSF of China (Grants 11171178 and 11271225), Science and Technology Project of High Schools of Shandong Province (Grant J12LI52) and Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JS completed the main part of this article, ZZ and FM corrected the main theorems. All authors read and approved the final manuscript.

Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.